Definition

Also denoted as

If $S$ is a singleton, that is: $S = \set x$, then we can (and usually do) write $G = \gen x$ instead of $G = \gen {\set x}$.

Also known as

This is also voiced:

$S$ is a generator of $\struct {G, \circ}$

$S$ generates $\struct {G, \circ}$

Some sources refer to such an $S$ as a set of generators of $G$, but this terminology is misleading, as it can be interpreted to mean that each of the elements of $S$ is itself a generator of $G$ independently of the other elements.

Other sources use the term generating set, which is less ambiguous.

Some sources use the notation $\operatorname {gp} \set S$ for the group generated by $S$.